1-Completions of a Poset

Home , Converse relation, Join and meet, Order embedding

Order (2013) 30:39–64 DOI 10.1007/s11083-011-9226-0

1-completions of a Poset

Mai Gehrke · Ramon Jansana · Alessandra Palmigiano

Received: 10 October 2010 / Accepted: 5 July 2011 / Published online: 27 July 2011 © The Author(s) 2011. This article is published with open access at Springerlink.com

Abstract A join-completion of a poset is a completion for which each element is obtainable as a supremum, or join, of elements from the original poset. It is well known that the join-completions of a poset are in one-to-one correspondence with the closure systems on the lattice of up-sets of the poset. A 1-completion of a poset is a completion for which, simultaneously, each element is obtainable as a join of meets of elements of the original poset and as a meet of joins of elements from the original poset. We show that 1-completions are in one-to-one correspondence with certain triples consisting of a closure system of down-sets of the poset, a closure system of up-sets of the poset, and a binary relation between these two systems. Certain 1-completions, which we call compact, may be described just by a collection of filters and a collection of ideals, taken as parameters. The compact 1-completions of a poset include its MacNeille completion and all its join- and all

The research of the second author has been partially supported by SGR2005-00083 research grant of the research funding agency AGAUR of the Generalitat de Catalunya and by the MTM2008-01139 research grant of the Spanish Ministry of Education and Science. The research of the third author has been supported by the VENI grant 639.031.726 of the Netherlands Organisation for Scientific Research (NWO). M. Gehrke (B) FNWI, IMAPP, Radboud Universiteit Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands e-mail: [email protected]

R. Jansana Dept. Lògica, Història i Filosofia, de la Ciència, Universitat de Barcelona, Montalegre, 6, 08001, Barcelona, Spain

A. Palmigiano FNWI, ILLC, Universiteit van Amsterdam, P.O. Box 94242, 1090 GE Amsterdam, The Netherlands 40 Order (2013) 30:39–64 its meet-completions. These completions also include the canonical extension of the given poset, a completion that encodes the topological dual of the poset when it has one. Finally, we use our parametric description of 1-completions to compare the canonical extension to other compact 1-completions identifying its relative merits.

Keywords Completions of a poset · Canonical extensions

1 Introduction

Embedding partially ordered sets in complete lattices is central to the solution of many problems. This is most famously exemplified by the construction of the real numbers from the rational numbers [6]. Dedekind’s construction, generalised to arbitrary posets by MacNeille [24], yields a completion for any poset P.The MacNeille completion of a poset is minimal in some sense but it does not have good algebraic preservation properties, e.g. it does not preserve homomorphisms, equational properties, and the like. For this purpose one would rather want a greatest completion, or in other words, a free completion. One-sided free completions exist and have indeed been used to great success, e.g. in domain theory. However, two- sided free completions (that is, completions that are free with respect to complete lattice homomorphisms rather than just with respect to complete join-morphisms or complete meet-morphisms) are well known not to exist—see e.g. [20, Section 4.7]. In [21, 22] Bjarni Jónsson and Alfred Tarski introduced a two-sided completion for Boolean algebras, called canonical extension, which is the free completion within the class of completely distributive (or equivalently, atomic) complete Boolean algebras. The canonical extension is closely related to duality theory and provides an algebraic approach to topological duality [15, 16]. As a completion, it has the virtue of preserving more equational properties than the MacNeille completion [13] and it has played a substantial role, in a more or less hidden way, in the semantic study of many logics including modal and intuitionistic logic [4, 7]. More recently, canonical extension has been generalised to encompass distributive lattices [14], lattices [12], and even posets [8] and the theory has been applied in logic and algebra, e.g. [1, 8, 15, 16]. Canonical extensions have the drawback that they do not preserve existing infinite meets and joins and, in the other direction, they do not add finite joins and meets in a free way. Depending on the application, e.g. fixed- point logics versus very weak fragments of logic [18], different completions may be appropriate. In this paper we study a class of two-sided completions encompassing both the MacNeille completion and the canonical extension. The identifying property of these completions is their placement in a meet-join complexity hierarchy for completions. This hierarchy was first brought to the attention of the first author by Keith Kearnes in a private communication and was subsequently studied in relation to canonical extension in [17]. The completions that we focus on here are called 1-completions and figure in the hierarchy as the completions for which each element is reachable by joins of meets and by meets of joins of elements from the original poset. Included among the 1-completions are the 0-completions which are the well-known join- completions as well as the 0-completions which are the meet-completions. The canonical extension of a poset is an example of a 1-completion which, in general, is Order (2013) 30:39–64 41 neither a 0-completion nor a 0-completion. Any 1-completion can be built from a collection of up-sets and a collection of down-sets of the original poset. However, possibly somewhat surprisingly, specifying collections of up-sets and down-sets is not in general sufficient for specifying a 1-completion. Our main result (Theorem 3.3) is a complete classification of the 1-completions of a poset P in terms of certain polarities (F, I, R) where F is a closure system of up-sets of P and I is a closure system of down-sets of P and R is a relation from F to I satisfying four simple conditions. The relation essentially specifies which meets of down-sets are below which joins of up-sets in the completion. Given a poset P, and closure systems F and I of up-sets and down-sets of P, respectively, there is always a smallest relation R from F to I satisfying the four conditions of Theorem 3.3. It is the relation of non-empty intersection, that is, for every F ∈ F and every I ∈ I,

FRI iff F ∩ I = ∅. (1)

In fact, even for collections of up-sets and down-sets of P which are not closure systems, one may consider the polarity obtained by this relation of non-empty intersection and thus reach a larger class of 1-completions using only this one relation. This is the idea behind what we call compact 1-completions and leads to notions of compactness and denseness which are parametric in collections F and I of up-sets and down-sets, respectively. We show that, for each choice of F and I containing the principal up-sets and the principal down-sets, respectively, there is (up to isomorphism) a unique completion of P which is (F, I)-compact and (F, I)- dense. Thus parametric compactness and denseness allow a study of 1-completions in which the relational component may be omitted in the sense that it is fixed to be the relation of non-intersection as given by Eq. 1. The class of parametrically compact and dense completions includes well-known completions such as the MacNeille completion, the canonical extension, and all join- and meet-completions of a given poset. However, we provide examples of posets for which it is not possible to describe all 1-completions as compact and dense with respect to some F and some I. Thus the relational component of our classification result is in general necessary. With the purpose of comparing canonical extension as defined in [8] to other completions which are (F, I)-compact and (F, I)-dense for some choice of F and I, we give conditions on F and I corresponding to various properties of the (F, I)-compact and (F, I)-dense completion of P, concentrating on properties which are central to applications in logic. The paper is organised as follows. In Section 2 we introduce 1-completions and show that MacNeille completions and Galois connections are central to the study of these completions. In Section 3 we briefly review the necessary preliminaries on Galois connections including the abstract characterisation of the lattice of Galois closed sets of a polarity as well as its construction as the MacNeille completion of a certain ‘intermediate’ structure. These tools then allow us to give a complete classification of all 1-completions of a given poset. In Section 4 we introduce what we call 1-polarities which are based on collections of up-sets and down-sets that are not necessarily closure systems. In Section 5 we study parametric compactness and in Section 6 we identify the properties required of F and I in order that the corresponding completion have various desirable properties. 42 Order (2013) 30:39–64

2 1-completions

We fix a poset P. A completion of P is an embedding of P inacompletelattice.Tobe precise, an order embedding is a map e: P → Q from a poset P into a poset Q so that, for every x, y ∈ P, we have x ≤ y if and only if e(x) ≤ e(y).Aposet extension of P is a pair (e, Q),whereQ is a partially ordered set and e: P → Q is an order embedding. A completion of P is a poset extension (e, C) of P such that C is a complete lattice. In order to lighten the notation we will generally identify P with e[P] so that e is the inclusion map, and thus we will talk about Q or C being an extension or a completion of P. Completion is a self-dual concept. However, in general, order properties and order theoretic results come in pairs of dual statements, since ≤ being an order implies that the converse relation, ≥, is an order as well. Throughout the present paper, we will take for granted that each definition and each result also yields a dual definition or result, indicating only the additional notation as needed. A subset X of P is a down-set of P provided that for every x ∈ X and y ∈ P we have that y ≤ x implies y ∈ X. For x ∈ P we denote by ↓x the down-set { y ∈ P | y ≤ x } and the least down-set that includes X ⊆ P will be denoted by ↓X. We denote the complete lattice of all down-sets of P with the inclusion order by D(P). Likewise, the up-set of x ∈ P will be denoted by ↑x, the least up-set containing X ⊆ P will be denoted by ↑X, and the complete lattice of all up-sets of P will be denoted by U(P). The restriction of a down- or up-set to a subposet Q will be denoted by a subscript ↓ =↓ ∩ on the arrow, e.g. Q X X Q. Given a completion C of P, we have the following fundamental associated Galois connection: : D( ) :↓ P C P C → I C I ↓x ∩ P ← x. : U( ) :↑ The order dual Galois connection is denoted C P C P. The fact that these are indeed Galois connections is easy to check. In the case of the second pair of maps, this means that for each up-set F ∈ U(P) and each element x ∈ C we have that ≤ ⊆↑ ( ) x C F if and only if F P x which is essentially the content of the definition of meet. Note that the same assignments above define Galois connections on the full power set of P, but the ↑ ↓ U( ) D( ) image of the maps P and P will always lie in P and P so that any completion of P gives rise to closure operators (given by the composition of the two Galois maps in each case) on U(P) and D(P), respectively. The Galois closed sets of the first Galois connection are: I ={ ∈ D( ) |↓ ( ) = }={ ∈ D( ) |∃ ∈ ↓ ( ) = } C I P P I I I P x C P x I C I( ) ={ ∈ | ↓ ( ) = }={ ∈ |∃ ∈ D( ) = }. C y C C P y y y C I P C I y

The corresponding notation for the second Galois connection is FC and F(C), respectively. We refer to the elements of IC (FC)astheC-normal ideals (C-normal Order (2013) 30:39–64 43 f ilters)ofP relative to the completion C. The elements of I(C) (F(C)) will be referred to as the ideal elements (f ilter elements) of the completion C.WealsocallI(C) (F(C)) the join-closure (meet-closure) of P in C. An extension Q of a poset P is join-dense provided each element of Q is the join of a collection of elements from P and a join-completion of P is a join-dense completion of P. It is clear that a completion C of P is a join-completion if and only if C = I(C) and that, for any completion C, the extension I(C) of P is a join-completion of P. The correspondence C → IC is in fact the well-known correspondence between closure systems on D(P) that contain the principal down-sets and join-completions of P [2]. Following Erné, [9], we will call a collection I of down-sets of P standard provided ↓x ∈ I for every x ∈ P, and dually for systems of up-sets. Thus the join- completions of a poset P are in one-to-one correspondence with the standard closure systems of down-sets of P and meet-completions are in one-to-one correspondence with standard closure systems of up-sets of P. An extension Q of P is called doubly dense provided it is both join- and meet- dense. It is well known that the MacNeille completion N (P) of a poset P is, up to isomorphism, the only completion of P that is doubly dense [3]. In this case the elements of the corresponding closure systems IN (P) and FN (P) are exactly what is usually known in the literature as the normal ideals and the normal filters of P. We may think of the completions of a poset as lying in a complexity hierarchy according to the (possibly transfinite) number of alternations of joins and meets one has to apply to generate the completion [17]. Thus the 0-completions of a poset are its join-completions, the 0-completions are its meet-completions, the 1- completions are those completions of P, every element of which is obtainable as a join of meets of elements from P, and so on. A completion is called a n-completion if it is both a n-andan-completion. Thus the MacNeille completion of a poset is its unique 0-completion. The 1-completions of P are those completions, each element of which may be obtained both as a meet of joins of elements of P and as a join of meets of elements of P. In the canonical extension literature, these completions have been called dense completions [12] and the canonical extension of any poset is such a completion [8]. Our purpose in this paper is to study the 1-completions of a poset. In this section we begin with a simple observation which allows us to see any 1-completion as the MacNeille completion of an extension that is the union of a join-completion and a meet-completion of the original poset.

Proposition 2.1 Let P be a poset. A completion C of P is a 1-completion of P if and only if C is the MacNeille completion of the poset F(C) ∪ I(C).

Proof This follows immediately from the fact that the MacNeille completion of a poset is uniquely determined as the doubly dense completion of the poset, along with the fact that the poset F(C) ∪ I(C) is join-dense in C if and only if F(C) is join-dense in C (since F(C) is always join-dense in I(C) because it contains P)andtheorder dual fact about meet-density of F(C) ∪ I(C) and I(C).

Let C be a 1-completion of P.ThenF(C) is a meet-completion of P and it corresponds to the standard closure system FC of up-sets of P. Dually, I(C) is a join-completion of P and corresponds to the standard closure system IC of 44 Order (2013) 30:39–64 down-sets of P. It is clear that isomorphic completions yield identical closure systems F and I. However, as the following example shows, it may happen that non-isomorphic 1-completions C and C of a given poset P satisfy FC = FC and IC = IC . Consequently, knowing FC and IC is not always sufficient for deter- mining C.

Example 2.2 Consider the poset P = ω ⊕ ω∂ , that is, the poset given by a countably infinite ascending chain under a countably infinite descending chain. The MacNeille completion of P is obtained by adjoining a point in between the chains, and the canonical extension of P is obtained by adjoining the two-element chain between the ∂ two infinite chains. So C1 = ω ⊕ 1 ⊕ ω is the MacNeille completion and C2 = ω ⊕ 2 ⊕ ω∂ is the canonical extension of P. Let us find the closure systems corresponding to these two 1-completions. It should be clear that in both cases the meet-closure of P adjoins one point to P. For C1 this is the unique point in the middle, while it is the top of the two-element chain for C2. However, in both cases the corresponding closure system F of up-sets is the set of all non-empty up-sets, which contains just one non-principal up-set, namely ω∂ . Similarly, the closure system I of all non-empty down-sets of P is the closure system of down-sets corresponding to the meet-closures of P in both C1 and C2.

3 A Classification of the 1-completions of a Poset

It is clear from Example 2.2 that more information is needed to specify a 1- completion of a poset than the associated standard closure systems of up-sets and of down-sets of P. In fact, what is needed is to know how I and F are glued together to form the poset F(C) ∪ I(C) in Proposition 2.1. The most convenient way to present this information is to switch from MacNeille completions to Galois closed sets of polarities. Our approach to polarities will closely follow the approach taken in [11], and we will use the notation used there. For further details and background on polarities, Galois connections, and their relation to MacNeille completion see also [5, Chapters 3 and 7] and [10]. A polarity is a triple (X, Y, R) where X and Y are non-empty sets and R is a binary relation from X to Y. Such a polarity gives rise to a Galois connection given by ()R : P(X) P(Y) : R() A → { y |∀x (x ∈ A ⇒ xRy) } { x |∀y (y ∈ B ⇒ xRy) }← B. The Galois closed subsets of X and of Y are, respectively, G(X, Y, R) = A ⊆ X | A = R(AR) = R B | B ⊆ Y , G(X, Y, R)R = B ⊆ Y | B = (R B)R) = AR | A ⊆ X . These are both topped intersection structures and the maps ()R and R()restrict to mutually inverse dual order-isomorphisms between these complete lattices. Order (2013) 30:39–64 45

There are natural maps from X and Y whose images are join- and meet-generating subsets of G(X, Y, R), respectively, [11, Proposition 2.10]. These maps are given by : X → G(X, Y, R)ϒ: Y → G(X, Y, R) x → R({ x }R) y → R{ y }. In particular, for every polarity (X, Y, R),thesetIm() ∪ Im(ϒ) is doubly dense in G(X, Y, R). As a consequence, G(X, Y, R) is (up to isomorphism) the MacNeille completion of Im() ∪ Im(ϒ). Therefore, the ordered set Im() ∪ Im(ϒ) is particularly important for under- standing the lattice G(X, Y, R). We will call the poset Im() ∪ Im(ϒ) the intermediate structure of G(X, Y, R) and denote it by Int(X, Y, R). We can build the intermediate structure directly from the polarity (X, Y, R). To this end, we first take the disjoint union of X and Y and equip it with the pullback ≤ of the order on Im() ∪ Im(ϒ) (inherited from G(X, Y, R)) along the map ϒ, thus obtaining a pre-ordered set pre-Int(X, Y, R). We then take the quotient of this pre-ordered set by ≤∩≥to obtain the poset Int(X, Y, R).Thatis,forz1, z2 ∈ pre-Int(X, Y, R) := X Y,welet

z1 ≤ z2 iff ϒ(z1) ⊆ ϒ(z2). The resulting pre-order is described in the following proposition [11, Proposition 2.7]:

Proposition 3.1 Let (X, Y, R) be a polarity and z1, z2 ∈ pre-Int(X, Y, R).

(1) if z1, z2 ∈ X, then z1 ≤ z2 if and only if (∀y ∈ Y)(z2 Ry ⇒ z1 Ry); (2) if z1, z2 ∈ Y, then z1 ≤ z2 if and only if (∀x ∈ X)(xRz1 ⇒ xRz2); (3) if z1 ∈ Xandz2 ∈ Y, then z1 ≤ z2 if and only if z1 Rz2; (4) if z1 ∈ Yandz2 ∈ X, then z1 ≤ z2 if and only if

(∀x ∈ X)(∀y ∈ Y)[(xRz1 & z2 Ry) ⇒ xRy].

Note that if x ∈ X ∩ Y, then there are two copies of x in the disjoint union X Y: one, call it x, as element of X, and another, x, as element of Y. Then, x ≤ x if and only if (x) ⊆ ϒ(x) if and only if xRx. Moreover, x ≤ x if and only if ϒ(x) ⊆ (x). The MacNeille completion may be treated axiomatically as the unique doubly dense completion of a poset. Such an axiomatic treatment is also possible for lattices of Galois closed sets of polarities. This fact was central to the paper [11] and is spelled out in the following theorem.

Theorem 3.2 Let (X, Y, R) be a polarity. Then G(X, Y, R) is the unique (up to isomorphism) complete lattice equipped with mappings : X → G(X, Y, R) and ϒ : Y → G(X, Y, R) so that the following properties hold: (1) For x ∈ Xandy∈ Y (x) ≤ ϒ(y) if and only if xRy; (2) G(X, Y, R) is join-generated by Im(); (3) G(X, Y, R) is meet-generated by Im(ϒ). 46 Order (2013) 30:39–64

Proof As outlined above, G(X, Y, R) with and ϒ satisfies the three conditions of the theorem. We need to prove uniqueness. To this end, suppose C is a complete lattice equipped with maps : X → C and ϒ : Y → C satisfying (1)–(3) with G(X, Y, R) replaced by C.LetInt(C) = Im() ∪ Im(ϒ) and consider the map ϒ : X Y → Int(C). Denoting the pre-order on pre-Int(X, Y, R) by ≤ and the order on C by ≤C, it is now straightforward to show that for every z1, z2 ∈ X Y,

z1 ≤ z2 ⇔ ϒ(z1) ≤C ϒ(z2).

For example, for z1, z2 ∈ X we have

z1 ≤ z2 ⇐⇒ ∀ y ∈ Y (z2 Ry ⇒ z1 Ry)

⇐⇒ ∀ y ∈ Y ((z2)≤Cϒ(y) ⇒ (z1)≤Cϒ(y))

⇐⇒ (z1) ≤C (z2)

⇐⇒ ϒ(z1) ≤C ϒ(z2). Note that the third equivalence above is due to the assumption that property (3) holds for C. It follows that Int(X, Y, R) is isomorphic to Int(C), which implies that G(X, Y, R) is isomorphic to C since they are the MacNeille completions of order- isomorphic posets.

Now we can state the following Galois connection variant of Proposition 2.1.

Corollary 3.3 Let P be a poset. A completion C of P is a 1-completion of P if and only if C is (isomorphic to) the lattice of Galois closed sets of the polarity (F(C), I(C), ≤F×I) where ≤F×I denotes the order of C restricted to F(C) × I(C).

Proof Suppose C is a 1-completion of P. The inclusion maps of F(C) and I(C) into C satisfy the conditions of the above theorem for the polarity (F(C), I(C), ≤F×I), ∼ where x ≤F×I y if and only if x ≤ y in C for x ∈ F(C) and y ∈ I(C). Thus C = G(F(C), I(C), ≤F×I). For the converse, a completion of the form G(F(C), I(C), ≤F×I) is clearly a 1- completion, since P is join-dense in I(C) and meet-dense in F(C) and their images in G(F(C), I(C), ≤F×I) are, respectively, meet- and join-dense.

We shall next obtain a description of all the 1-completions of a poset P in terms of polarities (F, I, R) where F is a standard closure system of up-sets and I a standard closure system of down-sets of P.

Theorem 3.4 Let P be a poset. There is a one-to-one correspondence between 1- completions of P and polarities (F, I, R) where (1) F is a standard closure system of up-sets of P; (2) I is a standard closure system of down-sets of P; (3) the relation R ⊆ F × I satisf ies the following four conditions: (Pol 1) ∀p ∈ P, x ∈ F (p ∈ x ⇐⇒ xRp); (Pol 2) ∀p ∈ P, y ∈ I (p ∈ y ⇐⇒ pRy); (Pol 3) ∀x, x ∈ F, y ∈ I (x ⊇ x Ry ⇒ xRy); Order (2013) 30:39–64 47

(Pol 4) ∀x ∈ F, y, y ∈ I (xRy ⊆ y ⇒ xRy). Here ↑pand↓p and p are identif ied for every p ∈ P.

Proof We have stated this theorem in terms of collections of up- and down-sets of P in order to make it clear that 1-completions are built from the original poset P. However, since standard closure systems F of up-sets of P are in one-to-one correspondence with meet-completions of P, and standard closure systems I of down-sets of P are in one-to-one correspondence with join-completions of P,we may as well work abstractly with polarities (K, O, R) such that (1) K is a meet-completion of P; (2) O is a join-completion of P; (3) the relation R ⊆ K × O satisfies the following four conditions:

(Pol 1) ∀p ∈ P, x ∈ K (x ≤K p ⇐⇒ xRp); (Pol 2) ∀p ∈ P, y ∈ O (p ≤O y ⇐⇒ pRy); (Pol 3) ∀x, x ∈ K ∀y ∈ O (x ≤K x Ry ⇒ xRy); (Pol 4) ∀x ∈ K ∀y, y ∈ O (xRy ≤O y ⇒ xRy ).

Given a 1-completion C of P,weletK = F(C), O = I(C),andR =≤C ∩ (K × O). Then it is clear that all the required properties hold and that C =∼ G(K, O, R). Conversely, given a polarity (K, O, R) satisfying the conditions (1)–(3) above, we let C = G(K, O, R). It suffices to show that (p) = ϒ(p) = p for each p ∈ P, ∼ ∼ that I(C) = K via : K → C,thatF(C) = O via ϒ : O → C,andthatR =≤C∩ (K × O). Let p ∈ P.Thenp ∈ K and p ∈ O. In order not to confuse these two copies of p (before we have shown that it is justified to do so), we will call the former pK and the latter pO.SincepK ≤K pK, it follows that pK RpO by property (Pol 1), and since the order from K to O in C = G(K, O, R) is R, we get pK ≤C pO.Inordertoshow pO ≤C pK, we use the join-density of K in C.Letx ∈ K with x ≤C pO.ThenxRpO, and thus x ≤K pK by property (Pol 1) again. From this, using the fact that K is join- dense in C, we get pO ≤C pK. This finishes the proof that pK = pO. From now on, we do not distinguish between the elements of P as they sit in K and as they sit in O. Let x, x ∈ K with x ≤K x ,andlety ∈ O. Then, by property (Pol 3), x Ry implies xRy. On the other hand, if ∀y ∈ O (x Ry ⇒ xRy), then, since P ⊆ O and by property (Pol 1), ∀p ∈ P (x ≤K p ⇒ x ≤K p), and since K is a meet-dense completion of P, it follows that x ≤K x . Thus we have: x ≤K x ⇐⇒ ∀ y ∈ O(x Ry ⇒ xRy) ⇐⇒ x ≤C x .

The second equivalence holds because O is meet-dense in C. This proves that K = F( ) ⊆ = embeds inC. We now show that K C .LetS P,andx K S. We show = that x C S. First note that x is a lower bound for S in C by property (Pol 1). Also, if z ∈ C is a lower bound of S, then, since K is join-dense in C, we have z = { ∈ | ≤ } ∈ ≤ ≤ ∈ C x K x z .Butifx K and x Cz,thenx C p for every p S and ≤ ∈ ≤ = ≤ thus x K p for each p S. Therefore x K K S x and then x C x. It follows = { ∈ | ≤ }≤ = that z C x K x z x and thus x C S. By the order dual argument it follows that O embeds in C and that O = I(C). Since in G(K, O, R), it is always true that R =≤C ∩ (F(C) × I(C)), we also get that R =≤C ∩ (K × O). 48 Order (2013) 30:39–64

4 1-polarities

In this section, we will consider polarities for which the corresponding collections F and I aren’t necessarily closure systems. In this way we are able to reach more 1-completions with a uniform choice of the relation in the polarity. We begin with some motivation. Given a poset P, a standard closure system F of up-sets of P and a standard closure system I of down-sets of P, it is easy to see that the set of relations R ⊆ F × I satisfying conditions (Pol 1)–(Pol 4) in (3) of Theorem 3.4 is non-empty and closed under arbitrary non-empty intersections. Therefore, the least relation Rl ⊆ F × I satisfying these conditions exists. This least relation is in fact the “non- empty intersection relation”, that is,

xRl y iff x ∩ y =∅.

It is straightforward to check that Rl satisfies (Pol 1)–(Pol 4). To see that it is the smallest relation that satisfies the above mentioned conditions, let S ⊆ F × I be any such relation. Let x ∈ F and y ∈ I be such that xRl y,andletp ∈ x ∩ y.Hencep ∈ y, which implies pSy by condition (Pol 2), and since x ⊇↑p, condition (Pol 3) implies that xSy. The canonical extension of a bounded lattice as defined in [12] is an example of a 1-completion which corresponds to a polarity endowed with this smallest relation: for every bounded lattice L, the canonical extension of L is the 1-completion of Galois closed sets of the polarity (F, I, Rl) where F is the closure system of the lattice filters of L and I the closure system of the lattice ideals of L. The MacNeille completion of a poset P corresponds, via Theorem 3.4, to the polarity (FN , IN , RN ) where FN consists of the standard closure system of all filters of P that are normal and IN consists of the standard closure system of all normal ideals of P. Here normal filters are the sets X ⊆ P that are equal to the set of upper bounds of the set of lower bounds of X, and order dually for ideals. In Example 2.2, we saw a lattice for which the MacNeille completion and the canonical extension were different but corresponded to the same closure systems of down-sets and up- sets. According to Theorem 3.4 then, in that example, the relation RN cannot be the ∂ ∂ relation of non-empty intersection. Indeed, ω RN ω even though ω ∩ ω =∅. The MacNeille completion is however traditionally built from a different polarity, namely, it is the 1-completion G(Fl, Il, Rl) of Galois closed sets of the polarity (Fl, Il, Rl) where Fl ={↑p | p ∈ P } is the set of principal filters and thus the least standard collection of filters, Il ={↓p | p ∈ P } is the set of all principal ideals, and Rl is the relation {↑p, ↓q|↑p ∩↓q =∅}of non-empty intersection. Note that this relation Rl between these (generally) non-closure systems satisfies (Pol 1)–(Pol 4). This example shows that it is important to consider completions of a poset P obtained as lattices of Galois closed sets of polarities (F, I, R) such that F is a standard collection of up-sets of P and I is a standard collection of down-sets of P,andR satisfies (Pol 1)–(Pol 4), even in cases where F and I are not necessarily closure systems. We call these polarities 1-polarities over P. Recall that, as with every polarity, for a 1-polarity (F, I, R), the lattice C = G(F, I, R) of Galois closed sets is isomorphic to the MacNeille completion of Im() ∪ Im(ϒ).SinceIm() ∪ Im(ϒ) is isomorphic to Int(F, I, R), C is the MacNeille completion of Int(F, I, R). Order (2013) 30:39–64 49

We unravel the content of Proposition 3.1 in order to describe the pre-order of the intermediate structure pre-Int(F, I, R) =F I, ≤ of a 1-polarity (F, I, R).

Proposition 4.1 Let P be a poset and (F, I, R) a 1-polarity over P. The following properties hold for the pre-order ≤ of pre-Int(F, I, R):

(1) if x, x ∈ F then x ≤ x if and only if x ⊆ x; (2) if y, y ∈ I then y ≤ y if and only if y ⊆ y; (3) if x ∈ F and y ∈ I then x ≤ y if and only if xRy; (4) if x ∈ F and y ∈ I then y ≤ x if and only if

∀p, q ∈ P [(p ∈ yandx q) implies p ≤ q].

Proof We use Proposition 3.1 and the properties (Pol 1)–(Pol 4) of 1-polarities. Condition (3) of the current proposition is precisely Proposition 3.1(3) and is thus true. To prove (1), note that by Proposition 3.1(1) we have x ≤ x in pre-Int(F, I, R) if and only if, for every y ∈ I,ifx Ry then xRy.Ifx ⊇ x and y ∈ I with x Ry then xRy by property (Pol 3) so that indeed x ≤ x in pre-Int(F, I, R). Conversely, if x ≤ x in pre-Int(F, I, R) and p ∈ x then by property (Pol 1) x Rp and thus x ≤ p in pre-Int(F, I, R). By transitivity of the pre-order on pre-Int(F, I, R), we obtain x ≤ p, and thus xRp; therefore p ∈ x by the reverse implication in property (Pol 1). The proof of (2) follows by order duality. In order to prove (4), first suppose that y ∈ I and x ∈ F with y ≤ x in pre- Int(F, I, R).Ifp, q ∈ P with p ∈ y and x q then pRy and xRq by conditions (Pol 1) and (Pol 2) and thus, by (3), p ≤ y ≤ x ≤ q. Hence by transitivity, p ≤ q. For the converse, suppose y x in pre-Int(F, I, R).SinceF is join-dense in pre- Int(F, I, R) there is x ∈ F with x ≤ y but x x. Now by condition (1) of the present proposition, there is q ∈ P with q ∈ x but q ∈ x. Now it follows by property (Pol 1) and (3) that x q in pre-Int(F, I, R). In combination with x ≤ y we obtain y q and thus by (2) y ⊆↓q.Thatis,thereisp ∈ y with p q.

Corollary 4.2 Let P be a poset and (F, I, R) a 1-polarity over P. Then the following properties are satisf ied:

(1) for each p ∈ P, we have (↑p) = ϒ(↓p); (2) : F → G(F, I, R) is an embedding of F with the reverse inclusion order into the f ilter elements of G(F, I, R); (3) ϒ : I → G(F, I, R) is an embedding of I into ideal elements of G(F, I, R); (4) the map P → G(F, I, R) given by p → (↑p) = ϒ(↓p) is a 1-completion of P.

Proof For property (1) we have (↑p) ≤ ϒ(↓p) by (3) of Proposition 4.1 combined with the fact that property (Pol 1) (or (Pol 2)) implies that pRp. We have (↑p) ≥ ϒ(↓p) by (4) of the above proposition. Properties (2) and (3) of the corollary are equivalent to (1) and (2) of the above proposition. Also by (4) in the above proposition we have x = { p ∈ P | p ∈ x } and y = { p ∈ P | p ∈ y } in G(F, I, R) for each x ∈ F and y ∈ I so that the map P → G(F, I, R) given by p → (↑p) = ϒ(↓p) yields a 1-completion of P. 50 Order (2013) 30:39–64

As we saw in Example 2.2 a non-principal filter and a non-principal ideal may be identified by the equivalence relation ≤∩≥coming from the pre-ordered set F I, ≤. All that can be said is that such identifications will happen when both (3) and (4) of Proposition 4.1 are satisfied for non-principal x and y. However, it will not happen for the filters and ideals of a 1-polarity based on the relation of non-empty intersection.

Corollary 4.3 Let P be a poset, F be a standard collection of f ilters of P, and I be a standard collection of ideals of P, and consider the 1-polarity (F, I, Rl) with Rl = {x, y∈F × I | x ∩ y =∅}. Then, for every x ∈ F and for every y ∈ I,

y ≤ x & x ≤ y ⇐⇒ (∃p ∈ P)(x =↑p & y =↓p).

Among the 1-polarities of a poset P, we have the ones with the additional property that F and I are closure systems. We will call these full 1-polarities. According to Theorem 3.4, every 1-polarity (F, I, R) of a poset P corresponds s s s to a unique full 1-polarity of P, which we will denote (F , I , R ). The following proposition specifies how to obtain this full polarity from the original polarity.

Proposition 4.4 Let P be a poset and (F, I, R) be a 1-polarity over P. Then

G(F, I, R) =∼ G(F s, Is, Rs) where F s is the closure under arbitrary intersections of F in U(P) and Is is the closure under arbitrary intersections of I in D(P),andRs is given by

xRs y ⇔ (∀x ∈ F)(∀y ∈ I)(x ⊇ x & y ⊆ y ⇒ x Ry).

Proof We will identify F with [F] and I with ϒ[I]. To prove the statement, we just need to check that F s and Is are the closure systems corresponding to the meet- closure and the join-closure of F and I in C = G(F, I, R), respectively. The fact that Rs is given as above follows directly from the fact that F and I are join- and meet-dense in C, respectively. = { ∈ | ∈ } We have seen in the proof of Corollary 4.2 that x C p P p x for each x ∈ F,andx ≤ p if and only p ∈ x. Thus, in the correspondence between elements of the meet-closure in a completion C and C-normal filters of P, each element of F corresponds to itself. Further, since, by definition, F s is the closure system generated by F, it is clear that the closure system of up-sets corresponding to the completion C contains F s. For the converse, let u ∈ C be in the meet-closure of P and let F = ↑u ∩ P be the corresponding C-normal filter of P. Then, for each p ∈ P with p ∈ F we have u p.SinceF is join-dense in C,thereisx ∈ F with x ≤ u and x p in C. Now x and u are both filter elements of C,sox ≤ u is equivalent to F ⊆ x. Moreover, x p corresponds to p ∈ x, and thus F = { x ∈ F | F ⊆ x } as required. The proof for I is order dual.

Note that, for a 1-polarity (F, I, Rl) where Rl is the relation of non-empty s (F s, Is, s) intersection, the relation Rl in the corresponding full 1-polarity Rl does not need to be the relation of non-empty intersection, as is illustrated in the following example. Order (2013) 30:39–64 51

Example 4.5 We consider the poset P = ω ⊕ ω∂ and its MacNeille C = ω ⊕ 1 ⊕ ω∂ as in Example 2.2. Then C = G(Fl, Il, Rl) where Fl is the set of principal filters of P and Il is the set of principal ideals of P,andRl is the relation of non-empty intersection. However, recall, from Example 2.2, that the closure systems of normal filters and normal ideals corresponding to the completion C are {↑p | p ∈ P }∪{ω∂ } and {↓p | p ∈ P }∪{ω }, respectively. Thus these are the closure systems F s and Is. s We also see that in this case Rl is different from the non-empty intersection relation F s Is (ω∂ ,ω)∈ s ω∂ ∩ ω =∅ from to as Rl and .

5 Parametric Compactness

In Section 4, we saw that for standard closure systems F and I of up-sets and of down-sets, respectively, of a poset P there always exists a least relation R so that (F, I, R) is a 1-polarity, namely, the relation Rl of non-empty intersection. It is not hard to see that, in general, for F and I standard collections of up-sets and down-sets, respectively, (F, I, Rl) is a 1-polarity and that Rl is the least relation R ⊆ F × I for which (F, I, R) is a 1-polarity. Note that by Proposition 4.1(3) and Corollary 4.2(4), we have that a 1- completion C = G(F, I, R) satisfies R = Rl if and only if C satisfies the following compactness property: (∀ ∈ F)(∀ ∈ I)( ≤ ⇐⇒ ∩ =∅). (F, I) x y C x C y x y ( -Comp)

As illustrated in Example 4.5, a completion C may be obtained both from a 1- polarity whose relational component is the relation of non-empty intersection and from one for which the relation isn’t that of non-empty intersection. This leads to the following notion of parametric compactness.

Definition 5.1 Let P be a poset and let F and I be standard collections of up-sets and of down-sets of P, respectively. A completion C of P is (F, I)-compact provided ((F, I)-Comp) holds.

While many 1-completions can be described as (F, I)-compact for some choice of F and I, there are posets that have 1-completions that are not (F, I)-compact for any choice of F and I. This is illustrated by the following example.

Example 5.2 Let P be the poset consisting of the disjoint union of a countably infinite increasing chain and a countably infinite decreasing chain. Let us denote by ω,≤ the increasing chain and by ω∂ , ≤∂ the decreasing chain. Note that ω is a non-principal down-set and ω∂ is a non-principal up-set in P.LetF ={↑p | p ∈ ∂ P }∪{ω } and I ={↓p | p ∈ P }∪{ω }. Consider the 1-completion C as specified in Fig. 1. Note that ω∂ is a completely join-irreducible element and ω is a completely meet-irreducible element in C. Recall that taking the MacNeille completion of a poset never creates new completely join- or completely meet-irreducible elements. Therefore, it must be true that for any choice of F and Ifor which C is an (F, I)-completion, ω∂ ∈ F and ω ∈ I.Since ω∂ = ω∂ ≤ ω = ω,andω∂ ∩ ω =∅, it follows that C is not (F, I)-compact for any choice of F and I that generate C as a 1-completion. 52 Order (2013) 30:39–64

Fig. 1 A 1-completion that is not compact for any choice of F and I

Next, we will show, under an appropriate denseness condition, that the notion of (F, I)-compact completion of P captures the completion G(F, I, Rl). Since this involves arguments about non-complete poset extensions such as the intermediate structure Int(F, I, Rl), we work in the more general setting of poset extensions as defined in Section 2. Accordingly, let P be a poset and Q be an extension of P that is not necessarily complete. We may consider the restrictions to Q of the Galois connections from Section 2, whose Galois closed sets are: F( ) := ∈ | = ∈ U( ) , Q a Q a Q F for some F P I( ) := ∈ | = ∈ D( ) . Q b Q b Q I for some I P

The elements of these sets will be called f ilter and ideal elements of Q, respectively. Further, the elements of

FQ := { ↑a ∩ P | a ∈ F(Q) },

IQ := { ↓b ∩ P | b ∈ I(Q) } are the Q-normal f ilters and the Q-normal ideals of P, respectively. The latter filters and ideals are of course respectively in one-to-one correspondence with the filter elements of Q and the ideal elements of Q. Moreover, let F be a standard collection of up-sets of P,andI be a standard collection of down-sets of P.ThenwecallF- f ilter elements of Q and I-ideal elements of Q the elements in the following two sets, respectively: FF ( ) := ∈ | = ∈ F , Q a Q a Q x for some x II ( ) := ∈ | = ∈ I . Q b Q b Q y for some y

Clearly FF (Q) ⊆ F(Q) and II (Q) ⊆ I(Q) for any F and I. Order (2013) 30:39–64 53

Definition 5.3 Let P be a poset, Q an extension of P and F a standard collection of up-sets of P and I a standard collection of down-sets of P. We say that Q is (F, I)- compatible provided F ⊆ FQ and I ⊆ IQ. This is equivalent to saying that : F FF ( ) :↑ Q Q P → x Q x ↑a ∩ P ← a are well-defined, mutually inverse, isomorphisms and similarly for : I II ( ) :↓. Q Q P ∈ F ∈ I Further, if Q x exists for each x and Q y exists for each y ,thenwesay that Q is (F, I)-compact provided ((F, I)-Comp) holds with C replaced by Q.

Proposition 5.4 Let F be a standard collection of up-sets and I be a standard ∈ F collection of down-sets of a poset P. If Q x exists for each x and Q yexistsfor each y ∈ I,andQisan(F, I)-compact extension of P, then Q is (F, I)-compatible. ∈ F ∈↑( ) ∩ ≤ ↓ (F, I) Proof Let x and p Q x P.Then Q x p in Q and thus, by - ∩↓ =∅ ∈ =↑( ) ∩ compactness, it follows that x p ,hencep x. Therefore, x Q x P and so x ∈ FQ. Similarly for y ∈ I.

The next proposition provides a useful characterisation of the (F, I)-compatible extensions that are (F, I)-compact.

Proposition 5.5 Let Q be a (F, I)-compatible extension of P. The following are equivalent: (1) Qis(F, I)-compact; (2) for every a ∈ FF (Q) and every b ∈ II (Q) if a ≤ bthena≤ p ≤ bforsome p ∈ P.

F I Proof Assume (1) is true. Let a∈ F (Q) and b ∈ I (Q) with a ≤ b. Then there are ∈ F ∈ I = = = ≤ = x and y such that a Q x and b Q y. Consequently Q x a b ∈ ∩ = ≤ ≤ = Q y and by compactness there exists p x y.Hencea Q x p Q y b and we have shown that (1) implies (2). Now assume that (2) is true. Let x ∈ F andy ∈ I.Thenby (F, I)-compatibility = = ≤ ∈ a Q x and b Q y exist. Suppose that Q x Q y.By(2),thereisp P ≤ ≤ (F, I) =↑ with a p b.SinceQ is a -compatible extension of P, we have x P a =↓ ∈ ∈ ∩ =∅ and y P b. Therefore, p x and p y and x y as required.

Corollary 5.6 Let P be a poset, and Q an (F, I)-compact and (F, I)-compatible ex- F I tension of P. Then the sub-poset F (Q) ∪ I (Q) of Q is isomorphic to Int(F, I, Rl), where xRl y if and only if x ∩ y =∅.

Proof It follows from the definition of (F, I)-compatible extension that the set of F-filter elements of Q is isomorphic to F with the reverse inclusion order and dually for I. By Proposition 5.5, it follows that for any F-filter element a and any 54 Order (2013) 30:39–64

I-ideal element b, we have a ≤ b if and only if a ≤ p ≤ b for some p ∈ P. Finally, for x ∈ F and y ∈ I, by the definition of join and meet, we have y ≤ x if and only if p ≤ q for every p ∈ y and every q ∈ x. All in all, we see that the order on F I F ∪ I is precisely the order of Int(F, I, Rl) as described in Proposition 4.1 via the corresponding pre-order on F I.

Let P be a poset and (F, I, R) be a 1-polarity. In the 1-completion C = G(F, I, R) the set FF (C) is join-dense and the set II (C) is meet-dense. This property will be called (F, I)-denseness, but we define it in the broader context of poset extensions.

Definition 5.7 Let P be a poset and let F and I respectively be a standard collection of up-sets and a standard collection of down-sets of P. We say that a poset extension Q of P is (F, I)-dense if FF (Q) is join-dense in Q and II (Q) is meet-dense in Q.

If Q is a poset extension of P which is an (F, I)-compatible extension, then the properties of (F, I)-compactness and (F, I)-denseness lift to the MacNeille completion of Q. This is because the MacNeille completion of any poset Q preserves all existing joins and all existing meets and because Q is both join- and meet-dense in its MacNeille completion. For every extension Q of a poset P,wewriteQ for the completion of P obtained by taking the MacNeille completion N (Q) of Q and composing the embedding from P to Q with the MacNeille embedding of Q into N (Q).

Proposition 5.8 For every poset P and every (F, I)-compatible extension Q of P:

(1) Qisa(F, I)-compatible completion of P; (2) if Q is (F, I)-compact then so is Q; (3) if Q is (F, I)-dense then so is Q.

Proof Statement (1) is true because FF (Q) = FF (Q) and II (Q) = II (Q),since existing meets and joins are preserved by the MacNeille completion. Statement (2) also follows from the fact that meets and joins from Qare preserved ∈ F ∈ I ≤ by MacNeille completion: let x and y be such that Q x Q y.AsQ is (F, I)-compatible, the meet and the join are taken in Q, and thus, by compactness of Q, x ∩ y =∅. In order to prove statement (3), letu ∈ Q.Since Q is join- and meet-dense in Q,thereareY, Z ⊆ Q such that u = Y = Z. Further, since Q is an (F, I)- ∈ ⊆ II ( ) dense extension of P, we have, for every y Y,some Ay Q and, for every ∈ ⊆ FF ( ) = = = z Z,someBz Q with y Q Ay and z Q Bz.Since Q Ay Q Ay = = = = and Q Bz Q Bz, it follows that x A B in Q where A y∈Y Ay and = B z∈Z Bz.

Definition 5.9 Let P be a poset and let F and I respectively be standard collections of up-sets and of down-sets of P. We say that a completion C of P is an (F, I)- completion provided it is (F, I)-compact and (F, I)-dense. Order (2013) 30:39–64 55

Note that all (F, I)-completions are necessarily (F, I)-compatible by Proposi- tion 5.4.

Theorem 5.10 For every poset P, any standard collection F of up-sets of P, and any standard collection I of down-sets of P, the completion G(F, I, Rl) is (up to isomorphism) the unique (F, I)-completion of P.

Proof Let C be an (F, I)-completion of P. By Corollary 5.6, we have that FF (C) ∪ I I (C) is isomorphic to Int(F, I, Rl). Further, by the (F, I)-denseness, any (F, I)- completion of P must be the MacNeille completion of FF ∪ II .

Corollary 5.11 The (F, I)-compatible extensions Q of P that are both (F, I)-dense and (F, I)-compact are, up to isomorphism, the posets satisfying Int(F, I, Rl) ⊆ Q ⊆ G(F, I, Rl).

Proof This corollary follows immediately from Corollary 5.6 and (F, I)-denseness.

We end the section with a discussion of the relation between our parametric notion of compactness and other notions of compactness in the literature. The standard non- parametric notion of compact completion of a poset is the following: A completion , ⊆ ≤ C of a poset P is compact if and only if for every XY P,if C X C Y then ⊆ ⊆ ≤ there are finite sets X X and Y Y such that C X C Y . As we will see, under suitable assumptions on the parameters F and I, the parametric notion of compactness is equivalent to (non-parametric) compactness. In the sequel we will use the following notation. We will respectively use F and I to refer to elements of F and I and X, Y to refer to arbitrary subsets of the poset. We say that a completion C ofP is weakly (F, I)-compact provided for every ∈ F ∈ I ≤ ⊆ ⊆ F and every I ,if C F C I then there are finite sets X F and Y I ≤ (F, I) such that C X C Y. Obviously if C is an -compact completion of P then it is weakly (F, I)-compact. The converse holds if F is in addition a family of down- directed sets and I a family of up-directed sets.

Proposition 5.12 For every standard collection F of down-directed up-sets of P and every standard collection I of up-directed down-sets of P, if C is a completion of P, then C is (F, I)-compact if and only if C is weakly (F, I)-compact.

Proof We only need to prove the implication from right to left. To this end assume C is a weakly (F, I)-compact completion of P and that F ∈ F and I ∈ I satisfy ≤ (F, I) C F C I.SinceC is a weakly -compact extension, there are finite sets ⊆ ⊆ ≤ X F and Y I with C X C Y. The down-directness of F implies that there is p ∈ F with p ≤ X and the up-directedness of I that there is q ∈ I such that ≤ ≤ ∈ ∩ ∩ =∅ C Y q. It follows that p q.Hence,q F I and so F I .

Lemma 5.13 Let F be a standard collection of up-sets of a poset P and let I be a standard collection of down-sets of P. If C is an (F, I)-compatible and (F, I)-dense completion of P, then for every X ⊆ P, = { ∈ F | ⊆ } (1) C X C F X F ; = { ∈ I | ⊆ }. (2) C X C I X I 56 Order (2013) 30:39–64 { ∈ Proof We only prove (1), since (2) follows by order duality. The inclusion C F F | ⊆ }≤ X F C X immediately follows from the fact that

X ⊆ { F ∈ F | X ⊆ F }. ≤ ∈ { ∈ F | ⊆ } In order to prove the reverse inequality, let z C Xand q F X F . ≤ (F, I) = { | α ∈ We will show that z q.SinceC is -dense, z C b α for some set b α F κ }⊆F (C). Then, since C is (F, I)-compatible, for every α ∈ κ there exists Fα ∈ F = =↑ ∈ α ∈ such that C Fα b α and Fα P b α. Note that then for every p X and every κ = ≤ ⊆ ∈↑ = , we have b α C Fα p. Therefore, since X P, we obtain that p P b α Fα is true for every α ∈ κ and every p ∈ X.Hence,X ⊆ Fα, and consequently q ∈ Fα = ↑ α ∈ κ = ≤ P b α, for every . Therefore z C b α q, as required.

Proposition 5.14 Let P be a poset, and let F be a standard collection of up-sets of P and I a standard collection of down-sets of P. If F and I are algebraic closure systems, then for every (F, I)-compatible and (F, I)-dense completion C of P, we have that C is weakly (F, I)-compact if and only if C is compact.

Proof Suppose C is a (F, I)-compatible and (F, I)-dense completion of P. Assume that C is weakly(F, I)-compact. In order to show that C is compact, let X, Y ⊆ ≤ ⊆ ⊆ P be such that C X C Y. We need to find finite sets X X and Y Y ≤ = { ∈ F | ⊆ } with C X C Y . To this end we consider the sets FX F X F = { ∈ I | ⊆ } F I ∈ F and IY I Y I .Since and are closure systems, we have FX ∈ I = ≤ = and IY . Moreover, applying Lemma 5.13 we obtain C FX C X C Y (F, I) ⊆ ⊆ C IY . Then, since Cis weakly -compact, there are finite Z FX and W IY ≤ F I ⊆ such that C Z C W.Since and are algebraic, there are finiteX X ⊆ ⊆ = { ∈ F | ⊆ } ⊆ = { ∈ I | and Y Y such that Z FX F X F and W IY I ⊆ } = ≤ ≤ ≤ = Y I . It follows that C X C FX C Z C W C IY C Y ,as required. That compactness implies weak (F, I)-compactness is obvious.

Propositions 5.12 and 5.14 imply that if F and I are algebraic closure systems of down-directed and up-directed sets respectively, then the notions of (F, I)- compactness, weak (F, I)-compactness and compactness are coextensive when they are applied to completions C of P which are (F, I)-compatible and (F, I)-dense.

6 Properties of (F, I)-completions

The canonical extension of a poset P [8]isthe(F, I)-completion of P where F is the collection of all down-directed up-sets and I is the collection of all up-directed down-sets. From the perspective of 1-completions, this choice for F and I is just one among many possible choices. In this section, we identify the properties of F and I that are needed in order to obtain a completion with various properties that are crucial in making canonical extensions work the way they do. The key properties we will consider are: restricted distributive laws, having enough completely join- and meet-irreducibles, preserving existing finite meets and joins, and commuting with formation of order dual as well as products. Throughout this section we assume P is a poset, F is a standard collection of up- sets of P,andI is a standard collection of down-sets of P.WeletP∗ denote the Order (2013) 30:39–64 57

(unique up to isomorphism) (F, I)-completion of P, and we assume that P ⊆ P∗. That is, P∗ is the unique (F, I)-compact and (F, I)-dense completion of P.

6.1 Restricted Distributive Laws

Definition 6.1 A completion C of P is said to satisfy the (F, I)-restricted distributive laws provided that for every collection X of non-empty down-directed sets of F-filter elements of C and for every collection Y of non-empty up-directed sets of I-ideal elements of C we have (1) { A | A ∈ X }= { f [X ]| f ∈ (X ) }, (2) { B | B ∈ Y }= { f [Y]| f ∈ (Y) }, where (X ) is the set of all choice functions of X ,thatismaps f : X → X such that f(A) ∈ A for every A ∈ X .

We will provide a condition on F and I that implies that P∗ satisfies the (F, I)- restricted distributive laws. We need a lemma.

Lemma 6.2 Let A be a down-directed set of F-f ilter elements and B an up-directed I = {↑ | ∈ }∈F = {↓ | ∈ } set of -ideal elements of P. If F P p p A and I Pq q B ∈ I,then

A ≤ B =⇒ ∃ p ∈ A ∃q ∈ Bp≤ q holds in P∗. ⊆ FF ( ∗) ∈ = ↑ Proof Since A P , for every p A, we have p P p. Therefore,

= {↑ | ∈ }= . A P p p A F Order dually, for every q ∈ B,

= {↓ | ∈ }= . B P q q B I Now assume that A ≤ B. Therefore F ≤ I.SinceP∗ is (F, I)-compact, there is r ∈ F ∩ I. But then, there are p ∈ A and q ∈ B with p ≤ r ≤ q.

Proposition 6.3 Suppose F and I are closed under non-empty directed unions. Then P∗ satisf ies the (F, I)-restricted distributive laws.

Proof We just prove one of the laws since the other follows by order duality. Let ∗ X be a collection of non-empty down-directed sets of F-filter elements of P .Since every f ∈ (X ) is a choice function for X it follows that A ≤ f (A) ∈ f [X ] for every A ∈ X . Hence, for every A ∈ X , we have A ≤ f [X ].Thisimpliesthat A | A ∈ X ≤ f [X ]| f ∈ (X ) . To prove the converse inequality, let x = { A | A ∈ X }.SinceP∗ is (F, I)- dense, we have I ∗ x = b ∈ I (P ) | x ≤ b . 58 Order (2013) 30:39–64 Therefore, in order to show that { f [X ]| f ∈ (X ) }≤x, it is enough to prove that for every b ∈ II (P∗) with x ≤ b there exists f ∈ (X ) satisfying that f[X ]≤ b.Letb ∈ II (P∗) be such that A | A ∈ X = x ≤ b. Then A ≤ b for every A ∈ X . Since every A ∈ X is down-directed and

{ ↑ | ∈ } ∈ F, P p p A by Lemma 6.2, we have that for every A ∈ X there exists pb ∈ A with pb ≤ b.Then → ∈ (X ) the assignment given by A pb defines a map fb with the property that fb [X ]≤b.

6.2 Completely Irreducible Elements

Recall that an element j of a complete lattice C is completely join-irreducible provided that, for every A ⊆ C,if j = A then j ∈ A.Orderdually,anelementm of a complete lattice C is completely meet-irreducible provided that, for every B ⊆ C, if m = B then m ∈ B.LetJ∞(C) denote the set of all completely join-irreducible elements of C and M∞(C) the set of all completely meet-irreducible elements of C. A complete lattice is perfect provided J∞(C) is join-dense in C and M∞(C) is meet- dense in C.

Proposition 6.4 (1) J∞(P∗) ⊆ FF (P∗), (2) M∞(P∗) ⊆ II (P∗).

Proof These two claims follow immediately from the assumption thatC is (F, I)- dense. Indeed, if x ∈ J∞(P∗),then(F, I)-denseness implies that x = A for some set A ⊆ FF (P∗), and since x is completely join-irreducible, it follows that x ∈ A; hence x ∈ FF (P∗). The second statement follows by order duality.

Proposition 6.5 If F and I are closed under unions of chains, then P∗ is a perfect lattice.

Proof In order to prove that every element of P∗ is a join of completely join- irreducibles, it suffices to prove that, if a ∈ FF (P∗) and b ∈ II (P∗) are such that ≤ ≤ ≤ ∈ ∞( ∗) ∈ F ∈ I a b,then j a and j bfor some j J P .LetFa and Ib with a = Fa and b = Ib . So, Fa ≤ Ib . Therefore, Fa ∩ Ib =∅.SinceF is closed under unions of ⊆-chains, by Zorn’s lemma, there exists someF ∈ F that is maximal F = ∩ =∅ among the elements in that are disjoint from Ib .Let j F.FromF Ib ≤ = ≤ = (F, I) ∩ = follows that j b,forif F j b Ib then, by -compactness, F Ib ∅ =↑ ={ ∈ | ≤ } = ⊆↑ . We claim that F P j p P j p .Since j F, we have F P j.If ↑ ∩ =∅ the inclusion is proper, then, by maximality, P j Ib , and so there exists some ∈ ≤ ≤ =↑ p P with j p b, a contradiction. Consequently, F P j. To finish the proof, we show that j is completely join-irreducible. By (F, I)- F ∗ denseness, it is enough to prove that if A ⊆ F (P ) is such that j = A,then j ∈ A. Suppose that j = a for every a ∈ A. Hence, since j = A implies that Order (2013) 30:39–64 59

≤ ∈ < =↑ a j for every a A, we obtain a j,andsoF P j Fa , which implies, by ∩ =∅ ∈ ∈ maximality, that Fa Ib . Hence for everya A there exists some pa Ib such that a ≤ pa . Therefore F = a = A ≤ Ib and so, by (F, I)-compactness, F ∩ Ib =∅, a contradiction.

We provide a characterisation of the completely join-irreducible elements when F and I are closed under unions of chains. The following concepts generalise notions introduced by Urquhart [25] for filters and ideals in lattices and used by Hartung [23]. The term optimal comes from [19].

Definition 6.6 A filter F ∈ F is said to be (F, I)-optimal if there exists some I ∈ I such that F is maximal in the set { G ∈ F | G ∩ I =∅}, and similarly we say that I ∈ I is (F, I)-optimal if there exists some G ∈ F such that I is maximal in the set { J ∈ I | J ∩ G =∅}.

Proposition 6.7 Assume that F and I are closed under unions of chains. For every F ∈ F and every I ∈ I with F ∩ I =∅, (1) there is some (F, I)-optimal G ∈ F with F ⊆ GandG∩ I =∅; (2) there is some (F, I)-optimal J ∈ I with I ⊆ JandF∩ J =∅.

Proof To prove (1), one argues as in the proof of the previous proposition to obtain a maximal element G in the set { G ∈ F | F ⊆ G, G ∩ I =∅}.ThenG is also maximal in { G ∈ F | G ∩ I =∅}. Therefore G is (F, I)-optimal. The statement (2) is proved order dually.

Proposition 6.8 Assume that F and I are closed under unions of chains. For every p ∈ P, every F ∈ F, and every I ∈ I, (1) if p ∈ F, there is an (F, I)-optimal G ∈ F with F ⊆ Gandp∈ G; (2) if p ∈ I, there is an (F, I)-optimal J ∈ I with I ⊆ Jandp∈ J.

Proof To prove (1), it is enough to consider the element ↓p of I which is disjoint from F, and apply Proposition 6.7, and to prove (2), to consider the element ↑p of F. Proposition 6.9 An F ∈ F is (F, I)-optimal if and only if F is completely join- irreducible in P∗. Dually, I ∈ I is (F, I)-optimal if and only if I is completely meet- irreducible in P∗.

Proof Assume that F ∈ F is optimal. So, let I ∈ I be such that F is maximal in { G ∈ F ∗ F | G ∩ I =∅}and let x = F ∈F (P ). In order to prove that x is completely join-irreducible, suppose that x = X for X ⊆ P∗.SinceFF (P∗) is join-dense in P∗, we may assume that X ⊆ FF (P∗). Now suppose that x ∈ X. Then for every x ∈ < ⊂↑ ∈ X we have x x. Thus, F Px for every x X. From the maximality of F in { ∈ F | ∩ =∅} ↑ ∩ =∅ ∈ G G I follows that Px I . So, we choose for every x X an ∈↑ ∩ = = ≤ ≤ ∩ =∅ element px Px I.Then F x X px I. Therefore, F I , contrary to our assumption. Thus, x ∈ X and we conclude that F is completely meet-irreducible. 60 Order (2013) 30:39–64

Suppose F is not (F, I)-optimal. Then, for each I ∈ I with F ∩ I =∅there is a ∈ F ∩ =∅ = { | ∈ I ∩ =∅} GI with F GI and GI I .Letu GI I with F I .We = ≤ will show that u F. Clearly u F since each GI contains F.Inordertoshow that F ≤ u,by(F, I)-denseness, it suffices to show that u ≤ I implies F ≤ I for each I ∈ I. By contraposition, suppose that F I.Then, certainly ∩ =∅ F ∩ =∅ (F, I) F I . Thus we have GI I and by -compactness it follows that ≤ GI I.Since GI u by definition of u we conclude that u I.Thatis, = = { | ∈ I ∩ =∅} F u GI I with F I .If F is completely join irreducible this would mean that F = GI for some I, which is a contradiction of (F, I)- compatibility since F = GI by assumption. The statement for completely meet-irreducibles follows by order duality.

6.3 Preservation of Finite Meets and Joins

Proposition 6.10 (1) The f inite joins existing in P are preserved in P∗ if f every I ∈ I is closed under existing f inite joins. (2) The f inite meets existing in P are preserved in P∗ if f every F ∈ F is closed under existing f inite meets.

Proof We just prove (1) as (2) follows by order duality. Let p, q ∈ P and suppose that p ∨ q exists in P. To say that p ∨ q is the supremum of p and q in P∗ is, by (F, I)-denseness, equivalent to saying that, for every y ∈ II (P∗), p ∨ q ≤ y if and only if y is a common upper bound of p and q.Butp ≤ y if and only if p ∈ I where I ∈ I is the ideal corresponding to y and similarly for q and p ∨ q. Thus we have that p ∨ q is the join of p and q in P∗ if and only if p ∈ I and q ∈ I ⇐⇒ p ∨ q ∈ I, is true for every I ∈ I, which is precisely the statement that every I ∈ I must be closed under all existing joins.

6.4 Dual Partial Orders

Let P be a poset, F be a standard collection of up-sets of P and I be a standard collection of down-sets of P.LetP∂ = (P, ≥) be the dual poset of P.Weset F(P∂ ) := I and I(P∂ ) := F.ThenF(P∂ ) is a standard collection of up-sets of P∂ and I(P∂ ) is a standard collection of down-sets of P∂ .Let(P∂ )∗ denote the (unique up to isomorphism) (F(P∂ ), I(P∂ ))-completion of P∂ .

Proposition 6.11 (1) (P∗)∂ is (up to isomorphism) the (F(P∂ ), I(P∂ ))-completion (P∂ )∗. ∂ (2) FF(P )((P∂ )∗) =∼ II (P∗); ∂ (3) II(P )((P∂ )∗) =∼ FF (P∗).

Proof The order dual of (F, I)-denseness is clearly (F(P∂ ), I(P∂ ))-denseness and the order dual of (F, I)-compactness is (F(P∂ ), I(P∂ ))-compactness. Thus (1) follows by uniqueness. Order (2013) 30:39–64 61

The properties (2) and (3) follow immediately from (1). We just prove (2). By (1) we can take P∂ → (P∗)∂ as the (F(P∂ ), I(P∂ ))-completion. For any a ∈ (P∂ )∗,we ∂ have that a ∈ FF(P )((P∂ )∗) if and only if

= = = a (P∂ )∗ F (P∗)∂ F P∗ F for some F ∈ F(P∂ ) = I. Conversely, for every b ∈ II (P∗), we have

= = b P∗ I (P∂ )∗ I

∂ for some I ∈ I = F(P∂ ),sob ∈ FF(P )((P∂ )∗).

6.5 Products

Let P1 and P2 be bounded posets. That is, both posets have a least element, which we will denote by 0, and a largest element, which we will denote by 1. Further, let Fi and Ii be a standard collection of up-sets and a standard collection of down-sets of Pi, respectively, for i = 1, 2.LetF = F(P1 × P2) and I = I(P1 × P2) be a standard collection of up-sets and a standard collection of down-sets of the poset P1 × P2, respectively. Recall that N (P) denotes the MacNeille completion of P.

Proposition 6.12 If

(i) F ={F1 × F2 | F1 ∈ F1 and F2 ∈ F2},and (ii) I ={I1 × I2 | I1 ∈ I1 and I2 ∈ I2}, then

∗ × ∗ (F, I) ( × )∗ (1) P1 P2 is (up to isomorphism) the -completion P1 P2 . F ∗ F ∗ F ∗ F (( × ) ) =∼ F 1 ( ) × F 2 ( ) (2) P1 P2 P1 P2 , I ∗ I ∗ I ∗ I (( × ) ) =∼ I 1 ( ) × I 2 ( ) (3) P1 P2 P1 P2 .

Proof (1) By the uniqueness of the (F, I)-completion, it is enough to show that × → ∗ × ∗ (F, I) (F, I) the product embedding P1 P2 P1 P2 is an -compact and -dense × (F, I) ( ,v)∈ ∗ × extension. Let us abbreviate P1 P2 as Q. For -denseness, let u P1 ∗ ∗ (F , I ) = , = P2;sincePi is an i i -dense extension of Pi for i 1 2, we get that u A1 F ∗ v = ⊆ F i ( ) and A2 for some Ai Pi . Notice that the existence of the 0 in Pi and the F =∅ ={( , ) | ∈ } standardness of i guarantee that we can take Ai .LetA a1 a2 ai Ai . Clearly, (u,v)= A. So in order to complete this part of the proof, it is enough to F F ∗ ⊆ F ( ) ∈ ⊆ F i ( ) = show that A Q .Fromai Ai Pi it follows that ai Fi for some ∈ F = × ∈ F = Fi i. Then consider F F1 F2. By assumption, F . Moreover, Q F ( ∗ π1[F], ∗ π2[F]) = ( ∗ F1, ∗ F2) = (a1, a2). The second part of denseness P1 P2 P1 P2 follows by order duality. 62 Order (2013) 30:39–64

For (F, I)-compactness, let F ∈ F and I ∈ I, then we have: ≤ ⇔ π [ ], π [ ] ≤ π [ ], π [ ] F I ∗ 1 F ∗ 2 F ∗ 1 I ∗ 2 I Q Q P1 P2 P1 P2

⇔ π [ ]≤ π [ ], = , ∗ i F ∗ i I i 1 2 Pi Pi

⇔ πi[F]∩πi[I] =∅, i = 1, 2 ( )

⇔ (∃ui)(ui ∈ πi[F]∩πi[I]), i = 1, 2

⇔ (∃(u1, u2))((u1, u2) ∈ F ∩ I) ( ) ⇔ F ∩ I =∅ ∗ (F , I ) The equivalence marked with ( ) is true because Pi is an i i -compact extension of Pi, and by our assumptions on F and I. The equivalence marked with ( )istrue because of our assumptions on F and I. The properties (2) and (3) follow immediately from (1). We just prove (2). By (1) × → ∗ × ∗ (F, I) we can take the product embedding P1 P2 P1 P2 as the -completion. ∗ F ∗ For every a ∈ (P × P ) , we have that a ∈ F ((P × P ) ) iff a = ∗ F = 1 2 1 2 (P1×P2) ∗× ∗ F1 × F2 for some F = F1 × F2 ∈ F,andsomeFi ∈ Fi,iffa = (a1, a2) for P1 P2 Fi ∗ ai = ∗ Fi ∈ F (P ). Pi i

Notice that the hypothesis of boundedness in the proposition above is needed: for instance, if P1 is the 2-element chain, P2 is the 2-element antichain, and the collections F (resp. I) are the principal up-sets (resp. down-sets) in the three posets, ∗ then conditions (i) and (ii) are clearly verified; however, (P1 × P2) = N (P1 × P2) is the lattice obtained by adding a top and a bottom to the disjoint union of two ∗ × ∗ = N ( ) × N ( ) 2-element chains, whereas P1 P2 P1 P2 is the Boolean algebra with three atoms. Let us now further assume that, for i = 1, 2, the collections Fi and Ii respectively consist of down-directed up-sets and of up-directed down-sets.

Proposition 6.13 If, for i = 1, 2:

(a) πi[F]∈Fi for every F ∈ F,andF1 × F2 ∈ F for every Fi ∈ Fi; (b) πi[I]∈Ii for every I ∈ I,andI1 × I2 ∈ I for every Ii ∈ Ii; then,

(i) F ={F1 × F2 | F1 ∈ F1 and F2 ∈ F2},and (ii) I ={I1 × I2 | I1 ∈ I1 and I2 ∈ I2}.

Proof We only prove (i). The right–to–left inclusion immediately follows by the assumption. To prove the converse inclusion, let F ∈ F; it is enough to show that F = π1[F]×π2[F]. Clearly, F is included in the product set. If (u,v)∈ π1[F]×π2[F], then (u, x2), (x1,v)∈ F for some xi ∈ Pi; the down-directedness of πi[Fi] implies that F is down-directed; hence, there exists some (u,v) ∈ F which is less than or equal to both (u, x2) and (x1,v).Hence,(u ,v) ≤ (u,v), which proves that (u,v)∈ F. Order (2013) 30:39–64 63

6.6 Choosing the Collections F and I

The above analysis shows that the key properties for F and I are: being standard; closure under the relevant lattice operations; closure under directed unions. For the uniform assignment P → (F(P), I(P)), the key properties are its compatibility with order duality and with products as specified in Propositions 6.11 and 6.13. Being standard guarantees that the original poset embeds in the completion. We have assumed that this is what we want in the present paper because we are talking about completions of posets, but in applications where P plays the role of some set of formal generators, as e.g. in presentations in domain theory, this may actually not be desirable. However, modulo taking a quotient of P, the results here should go through to the setting of domain theory. Closure under the lattice operations may in fact be desirable in some cases but not in others. In the setting of lattices, we want lattice completions, so the closure under the lattice operations is key to this setting. In some other situations we may want to add finite joins or meets freely. This was for example the case in our work [18] where we used the results from the present paper to develop a notion of canonical extension for logics, as these are treated in Abstract Algebraic Logic. Having enough completely join- and completely meet- irreducible elements is essential if the purpose is to develop relational semantics on the basis of the extensions. Applications to logic crucially involve operations that reverse order in some coordinates. This is why having extensions that commute with order reversal is important. Finally, it is crucial in the theory of canonical extensions that the canonical extension of a Boolean product is the full product of canonical extensions. This latter compatibility certainly includes compatibility with respect to finite products, which is necessary to being able to treat n-ary operations uniformly. There are further reasons for wanting compatibility with products: in the lattice setting, the powerful algebraic version of the Fine-van Benthem–Goldblatt theorem, which guarantees inter alia the canonicity of finitely generated varieties, relies heavily on the compatibility of extensions with Boolean products. Note that, starting from the principal up-sets and down-sets and closing under directed unions gives precisely the usual notions of filters and ideals for posets, namely down-directed up-sets and up-directed down-sets, respectively. This minimal choice with respect to these two properties also has all the other properties listed, and thus it is, also from the point of view of the present analysis, a natural uniform choice for F and I. However, the benefit of the classification theorem in Section 3 and of the results on compact completions is that they provide a parametric class of completions that may contain the completions best suited in many different situations.

Acknowledgement We would like to thank the anonymous referee for many detailed and thought- ful suggestions, which very much improved the exposition and our use of the English language.

Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

1. Almeida, A.: Canonical extensions and relational representations of lattices with negation. Stud. Log. 91(2), 171–199 (2009) 64 Order (2013) 30:39–64

2. Banaschewski, B.: Hüllensysteme und Erweiterung von Quasi-Ordnungen. Z. Math. Log. Grundl. Math. 2, 117–130 (1956) 3. Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeille completion. Arch. Math. (Basel) 18, 369–377 (1967) 4. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press (2001) 5. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press (2002) 6. Dedekind, R.: Stetigkeit und Irrationale Zahlen, Authorised Translation Entitled Essays in the Theory of Numbers. Chicago Open Court Publisher (1901) 7. Dunn, J.M., Hardegree, G.M.: Algebraic Methods in Philosophical Logic. Oxford University Press, New York (2001) 8. Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symb. Log. 70(3), 713–740 (2005) 9. Erné, M.: Adjunctions and standard constructions for partially ordered sets. Contrib. Gen. Algebra 2, 77–106 (1983) 10. Erné, M.: Adjunctions and Galois connections: origins, history and development. In: Denecke, K., et al. (eds.) Galois Connections and Applications, pp. 1–138. Kluwer, Boston, MA (2004) 11. Gehrke, M.: Generalized Kripke frames. Stud. Log. 84, 241–275 (2006) 12. Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238, 345–371 (2001) 13. Gehrke, M., Harding, J., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005) 14. Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Jpn. 40, 207–215 (1994) 15. Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005) 16. Gehrke, M., Priestley, H.A.: Duality for double quasioperator algebras via their canonical extensions. Stud. Log. 68, 31–68 (2007) 17. Gehrke, M., Priestley, H.A.: Canonical extensions and completions of posets and lattices. Rep. Math. Log. 48, 133–152 (2008) 18. Gehrke, M., Jansana, R., Palmigiano, A.: Canonical extensions for congruential logics with the deduction theorem. Ann. Pure Appl. Logic 161, 1502–1519 (2010) 19. Haim, M.: Duality for lattices with operators: a modal logic approach. Master Dissertation MoL2000-02, ILLC. http://www.illc.uva.nl/Publications/reportlist.php?Series=MoL (2000) 20. Johnstone, P.T.: Stone Spaces. Cambridge University Press (1982) 21. Jónsson, B., Tarski, A.: Boolean algebras with operators, I. Am. J. Math. 73, 891–939 (1951) 22. Jónsson, B., Tarski, A.: Boolean algebras with operators, II. Am. J. Math. 74, 127–162 (1952) 23. Hartung, G.: A topological representation of lattices. Algebra Univers. 29, 273–299 (1992) 24. MacNeille, H.M.: Partially ordered sets. Trans. Am. Math. Soc. 42, 416–460 (1937) 25. Urquhart, A.: A topological representation theory for lattices. Algebra Univers. 8, 45–58 (1978)